Discrete Lagrange Problems with Constraints Valued in a Lie Group

2021/06/03 by Chacón, Pablo M., Fernández, Antonio, García, Pedro L.
#2020 MSC. Primary: 58A20 #49N99 #53A70 #53Z05 #58E30 #Differential Geometry (math.DG) #FOS: Mathematics #Secondary :57R15

paper · doi:10.48550/arxiv.2106.01995

Abstract

The Lagrange problem is established in the discrete field theory subject to constraints with values in a Lie group. For the admissible sections that satisfy a certain regularity condition, we prove that the critical sections of such problems are the solutions of a canonically unconstrained variational problem associated with the Lagrange problem (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established. The whole theory is applied to the Euler-Poincaré reduction in the discrete field theory, concluding as an illustration with the remarkable example of the harmonic maps of the discrete plane in the Lie group SO(n).

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